Given that,
A person invested a total amount of $ \text{Rs.} \; 15 \; \text{lakh} .$
A part of it he invested in a fixed deposit, and earns $6\%$ annual interest.
Let $y$ lakh be the amount invested in a fixed deposit.
So, $ \text{Principle} = y, \; \text{Rate} = 6\%$
The remaining invested amount $ = ( 15 – y )$
The remaining amount was invested in two deposits in the ratio $2:1,$ earning annual interest at the rates of $4\%$ and $3\%,$ respectively.
- First deposit $: \text{Principle} = \frac{2}{3} (15 – y), \; \text{Rate} = 4\% $
- Second deposit $: \text{Principle} = \frac{1}{3} (15 – y), \; \text{Rate} = 3\%$
The total annual interest income $ = \text{Rs.} 76000 = \frac{76000}{100000} \; \text{(lakh)} = .76 \; \text{lakh}$
We know that$,\boxed{\text{Annual interest income} \propto \left(\text{Principle} \times \text{Rate}\right)}$
Now$,6\% \times y + \frac{2}{3}(15-y)4\% + \frac{1}{3}(15-y)3\% = .76$
$ \Rightarrow \frac{6y}{100} + \frac{8}{300}(15-y) + \frac{1}{100}(15-y) = .76 $
$ \Rightarrow \frac { 18y + 8(15-y) + 3(15-y)}{300} = .76 $
$ \Rightarrow 18y + 120 – 8y + 45 – 3y = \frac {76 \times 300}{100} $
$\Rightarrow 18y + 165 – 11y = 228 $
$ \Rightarrow 7y = 228 – 165 $
$ \Rightarrow 7y = 63 $
$ \Rightarrow \boxed{y = 9}$
$ \therefore$ The amount invested in the fixed deposit $ = 9 \; \text{lakh}.$
Correct Answer $: 9$